
Total Variation admits another formulation as the dual norm of \({\cdot }_\infty \)  (it is an example of an integral probability metric [1]}).
It reads
\((|) = \sup _{{_\infty \le 1} \int _(-),so that TV induces the dual topology induced by the strong topology on ().Thus , and more generally Csiszár divergences with \phi ^{-1}(\lbrace 0\rbrace )=\lbrace 1\rbrace  (see~\cite [Corollary 2.9]{liero2015optimal}), induce a stronger topology than the weak^* convergence, i.e. \phi (_n|)\rightarrow 0 implies _n\rightharpoonup , but the converse does not hold.For instance take x_n\rightarrow y but x_n \ne y, one has _{x_n}\rightharpoonup _y, but (_{x_n}|_y)=+\infty  and (_{x_n}|_y)=2.Csiszár divergences integrate pointwise comparison of measures \tfrac{}{}(x) at any x\in , but ignore the proximity between positions (x,y), hence their incapacity to metrize the weak^* topology.}\) 
